ENGG 330

Class 2

Concepts, Definitions, and Basic Properties

Quiz

• What is the difference between– Stem & Plot– How do I specify a discrete sample space from

0 to 10– How do I multiply a scalar times a matrix– How do I express e3[n]

Remember

• Real world signals are very complex

• Can’t hope to model them

• Can model simple signals

• Can tell a lot about systems with simple signals

• Can model complex signals with, dare I say, transformations of simple signals

Transformations of the Independent Variable

• Example Transformations

• Periodic Signals

• Even and Odd Signals

Transformations of Signals

• A central concept is transforming a signal by the system– An audio system transforms the signal from a

tape deck

Example Transformations

• Time Shift – Radar, Sonar, Seismic– x[n-n0] & x(t-t0)

• Notice a difference? n for D-T, t for C-T

– Delayed if t0 positive, Advanced if t0 negative

• Time Reversal – tape played backwards– x[n] becomes x[-n] by reflection about n = 0

• Time Scaling – tape played slower/faster– x(t), x(2t), x(t/2)

Time Shift

t0 < 0 so x(t-t0) is an advanced version of x(t)

Time Reversal

Time Scaling

?

What does x(t+1) look like?

When t = -2 t+1 = -1 what is x(t) at –1? 0 When t = -1 t+1 = 0 what is x(t) at 0? 1When t = 0 t+1 = 1 what is x(t) at 1? 1When t = 1 t+1 = 2 what is x(t) at 2? 0

Th e other way – t + 1+1 advanced in time

Given x(t) what would x(t-1) look like?

?

What does x(-t+1) look like?

When t = -1 -t+1 = 2 what is x(t) at 2? 0When t = 0 -t+1 = 1 what is x(t) at 1? 1When t = 1 -t+1 = 0 what is x(t) at 0? 1When t = 2 -t+1 = -1 what is x(t) at –1? 0

The other wayx(-t + 1)

Apply the +1 time shift

Apply the –t reflection about the y axis

?

What does x(3 /2 t) look like?

When t = -1 3t/2 = -3/2 what is x(t) at -3/2? 0When t = 0 3t/2 = 0 what is x(t) at 0? 1When t = 1 3t/2 = 3/2 what is x(t) at 3/2? ?When t = 2/3 3t/2 = 1 what is x(t) at 1? 1Why 2/3?What is the next t that should be evaluated?4/3 why?

?

What does look like?

Next apply the 3t/2 and compress the signal

First apply the +1 and advance the signal

Signal Transformations

• X(at + b) where a and b are given numbers– Linearly Stretched if |a| < 1– Linearly Compressed if |a| > 1– Reversed if a < 0– Shifted in time if b is nonzero

• Advanced in time if b > 0

• Delayed in time if b < 0

• But watch out for x(-2t/3 + 1)

Periodic Signals

• x(t) = x(t + T) x(t) periodic with period T

• x[n] = x[n + N] periodic with period N

• Fundamental period T or N

• Aperiodic

Even and Odd Signals

• Even signals – x(-t) = x(t)– x[-n] = x[n]

• Odd signals – x(-t) = -x(t)– x[-n] = -x[n]– Must be 0 at t = 0 or n = 0

• Any signal can be broken into a sum of two signals on even and one odd– Ev{x(t)} = ½[x(t) + x(-t)]– Od{x(t)} = ½[x(t) – x(-t)]

Exponential and Sinusoidal Signals

• C-T Complex Exponential and Sinusoidal Signals

• D-T Complex Exponential and Sinusoidal Signals

• Periodicity Properties of D-T Complex Exponentials

C-T Complex Exponential and Sinusoidal Signals

• x(t) = Ceat where C and a are complex numbers– Complex number

• a + jb – rectangular form

• Rejθ – polar form

• Depending on Values of C and a Complex Exponentials exhibit different characteristics– Real Exponential Signals– Periodic Complex Exponential and Sinusoidal Signals– General Complex Exponential Signals

Real Exponential Signals• If C and a are real

– x(t) = Ceat then called real exponential

• If a is positive x(t) is a growing exponential• If a is negative x(t) is a decaying exponential• If a 0 x(t) is a constant

– That depends upon the value of C

• Use MATLAB to plot – e2n, e-2n , e0n , 3e0n

Periodic Complex Exponential and Sinusoidal Signals

• If a is purely imaginary– x(t) is then periodic

• x(t) = ejw0t – Plot via MATLAB

• ? j is needed to make a imaginary

• a closely related signal is Sinusoid

General Complex Exponential Signals

• Most general case of complex exponential– Can be expressed in terms of the two cases we

have examined so far

Periodicity Properties of D-T Complex Exponentials

Unit Impulse and Unit Step Functions

• D-T Unit Impulse and Unit Step Functions

• C-T Unit Impulse and Unit Step Functions

C-T & D-T Systems

• Simple Examples

Basic System Properties

• Memory

• Inverse

• Causality

• Stability

• Time Invariance

• Linearity

Memory

• Memoryless output for each value of independent variable is dependent on the input at only that same time

• Memoryless– y(t) = x(t), y[n]= 2x[n] – x2[2n]

• Memory– Y[n] = Σx[k], y[n] = x[n-1]

Inverse

• Invertible if distinct inputs lead to distinct outputs

• Think of an encoding system– It must be invertible

• Think of a JPEG compression system– It isn’t invertible

Causality

• A system is causal if the output at any time depends on values of the input at only present and past times.

• See Fowler Note Set 5 System Properties

Stability

• If the input to a stable system is bounded the the output must also be bounded– Balanced stick

• Slight push is bounded

• Is the output bounded

Time Invariance

• See Fowler Note Set 5 System Properties

Linearity

• See Fowler Note Set 5 System Properties

Assignment

• Read Chapter 1 of Oppenheim– Generate math questions for Dr. Olson

• Buck– Section 1.2 a, b, c, d– Section 1.3 a, b, c– Section 1.4 a, b

• Turn in .m files– All plots/stems need titles and xy labels– Answers to questions documented in .m file with

references to plots/stems